Stretching yardstick

Turkish (my native language) and English. In Turkey, a lot of teachers speak in Turkish but write in English.

I got my PhD in mathematics almost 2 years ago and now I am a postdoctoral fellow.

I had a problem with dilation on the Euclidean plane. If all the coordinates of the points of the plane are multiplied by a, say positive, constant, then all the lengths of line segments on the plane are also multiplied by this constant. It was only obvious for line segments that touch the origin on one side. I visualised the process as pushing the points away from the origin. I could also see that this would result in pushing all points away from each other. I just could not see that the “amount of pushing” in all directions was the same. Somehow, I had this idea that if all the distances are multiplied by the same number, the dilation cannot have a center, that is, a fixed point.

It is remarkable how easy I can relate to that story — at about the same age I had a similar uncertainty of my feelings about the model of expanding universe: no matter from where you look at the universe, you see yourself at the centre of dilation, and the coefficient is the same every time.

Our geometric intuition is firmly rooted in the assumption that the unit of length is constant. There is a class of mathematical conundra exploiting this weakness of our intuition. For example,

A 1 meter long rubber tape is placed on the floor and one end nailed; another end is being pulled away with constant speed 1 cm per minute, so that the tape stretches. A snail started to crawl at speed 1 cm per minute from the fixed end towards the moving end. Will it ever reach it?

And if you amused by my use of “conundra” as a plural of “conundrum“, read a discussion in The Guardian‘s “Notes and Queries“.

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Good puzzle! My conclusion is that the snail would indeed reach the end, because each point along the rubber band is moving away at a rate proportional to its distance along the tape. So when the snail is at a distance x (between 0 and 1) from the fixed end, it is moving at a rate of 1 cm per minute, but the moving end is moving away at a rate of 1-x cm per minute. So the distance between the snail and the moving end is shrinking. (I’m going to have to think about this carefully, though, because some aspects about the problem are still making me uneasy. You’re right about it being counterintuitive!)